Use a dividend discount model with several growth stages

CFA® Level I · Equities · Discounted Cash Flow (DCF) and Growth Models · EQ.1.c

The idea

A multi-stage model discounts an explicit forecast of dividends over a growth phase and then applies a terminal value, usually the constant growth formula, to the point at which growth has settled. The terminal value often dominates the answer, in many cases supplying three quarters or more of the total depending on the horizon, growth and discount rate, so the terminal growth rate deserves more scrutiny than the detailed near-term forecast that consumes most of the effort. A two-stage model with an abrupt drop in growth is a simplification; an H-model fading growth linearly is often the more plausible shape.

Formulas

Multi-stage dividend discount model

V₀ = Σ [Dₜ ÷ (1 + r)ᵗ] + terminal value ÷ (1 + r)ⁿ

Dₜ: the dividend forecast individually for each year of the first stage
r: the required return
t: the year a dividend falls in, 1 through n
n: the last year of the first stage
terminal value: the Gordon value of everything after year n, as at year n

A company growing fast now and expected to settle later. Two errors to avoid: the terminal value is struck at year n and so is discounted n years, not n + 1, and it uses the dividend of year n + 1.

H model for a fading growth rate

V₀ = D₀ × (1 + gL) ÷ (r − gL) + D₀ × H × (gS − gL) ÷ (r − gL)

V₀: the value of the share today
D₀: the dividend just paid
gL: the long-run growth rate, after the fade
gS: the starting growth rate, before the fade
H: half the length of the fade period, in years
r: the required return

Growth starts high and falls in a straight line to the long-run rate. The first part is the Gordon value at the long-run rate and the second part is the extra value from the fade. It is an approximation.

A practice question

From the question bank, with the reason each wrong option is wrong.

What does a multi-stage dividend discount model add to the constant growth model?

An initial period of unusual growth before a steady rate - correct
A separate discount rate for each forecast yearWhy not: Varying the discount rate is possible but is a separate refinement. What the multi-stage model varies is the growth rate.
Removal of the need for a terminal valueWhy not: The multi-stage model still needs a terminal value: once the explicit period ends, the constant growth formula supplies the value of all later dividends.

The basic model, Value = D1 / (r - g), assumes one steady growth rate for ever. A multi-stage model adds an explicit forecast period in which dividends are forecast year by year, for a time when growth is unusually high or low. After that period, growth settles at a lasting, sustainable rate. Formula: Value = sum of discounted dividends in the explicit period + discounted terminal value. The terminal value, found with the constant-growth formula, covers all dividends after the explicit period, so it is still needed.

A harder one

Kenji Watanabe values Halden Foods with a two-stage dividend discount model. The present value of the dividends in the five-year growth phase is $7.40 per share, and the present value of the terminal value, calculated with the constant growth formula at the end of year five, is $26.60 per share. He has limited time to stress-test his inputs. Which input should receive the most scrutiny, given how the value is built up?

Year-one dividend, because it is discounted the leastWhy not: The year-one dividend is only one part of a $7.40 total that is itself just 22% of the value, so even a sizeable error in it barely moves the answer.
Detailed annual forecasts, because they took most effortWhy not: Effort spent is not a guide to impact. The five explicit years together supply only 22% of the value, so the forecasts that consumed most of the work matter far less than the terminal assumption.
Terminal growth rate, because it drives 78% of the value - correct

Add the pieces: Total value = PV of dividends + PV of terminal value = 7.40 + 26.60 = $34.00. The terminal share is 26.60 / 34.00 = 78%, so the five years of dividends are only 22%. The terminal value comes from: Next dividend / (r - g), so it depends heavily on the terminal growth rate g. The input that drives 78% of the value is the one to test hardest. The year-one dividend and the detailed annual forecasts matter much less, even though they look like hard work.

The hardest tier

In a two-stage dividend discount model, why is an abrupt drop in growth at the end of stage one a simplification worth questioning?

Because it makes the model harder to computeWhy not: A step is the easier case to compute. The gradual fade is the extra work.
Because advantage usually erodes gradually - correct
Because it always overstates the valueWhy not: It can overstate or understate depending on where the step is placed. The objection is that it misdescribes the shape.

A two-stage model has fast growth for some years, then drops straight to a lower steady rate. That sudden drop is a simplification. In real life a company's advantage over competitors, such as a strong product, fades slowly as rivals catch up, not on one particular date. Growth therefore tends to slide down gradually. An H-model uses a straight-line fade instead. The step does not make the sum harder, and it can overstate or understate value depending on where it is placed. The objection is that it gets the shape wrong.

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